The aim of Study 2 was to determine the effect of physical objects on children‘s partitioning strategies by comparing children‘s performance using cubes with no materials. It was predicted that by providing the initial amount of cubes, the procedural task demands would be reduced and children would identify more correct solutions with cubes than without. However, the study also aimed to compare the strategies used, which was achieved by developing a means to code the solutions given. By comparing the type of solutions given using cubes or using no materials, this study tested whether physically manipulating representations encouraged or discouraged the use of strategies that relate successive solutions (compensation and commutative strategies).
3.2 Method
3.2.1 Design
A within subjects design was used with Condition (Physical/No Materials) as the within subjects independent variable. The primary dependent variable was the number of correct solutions. These solutions were then coded according to a coding scheme developed in this study, thereby creating a further dependent measure: frequency of solutions in particular strategy categories.
3.2.2 Participants
Thirty two children took part in this study (17 girls and 15 boys, range 68 to 82 months;
M=74.2; SD=3.86 months). Children were from two Year 1 groups in a local school in
152 Nottingham, and their parents had signed and returned a consent form asking if they would like their child to take part in the study (56% response). This school was chosen from previous research with the university but children had not participated in a previous study in this research. The school is a larger than average primary school, with 345 pupils and situated in a suburb of Nottingham that is recognised as having a high social, educational and economic level. This is reflected in the very small proportion (2%) of children that receive free school meals.
In this study, all but one child had English as their first language and one child was reported as having special needs. These children were competent in the task and were included in the analysis. The school requested that no personal information including individual ability levels be taken.
3.2.3 Materials and Procedure
Sessions took place individually on a table in the corridor outside the class. They were held during lessons when noise levels in this area were acceptably low, and lasted between five and ten minutes. Children were presented with two partitioning problems:
partitioning 6 followed by partitioning 7, always in this order. The order of condition (Physical/No Materials) was counterbalanced, changing for each child in turn. The order of children reflected an alphabetic class list, which made it easier for the class teacher to know who was next and was deemed sufficiently randomised for this within subjects design.
As children had only briefly been introduced to the interviewer by the teacher in class, the interviewer spent up to a minute at the start of the session chatting informally
153 to help each child relax. The interviewer would then explain that the purpose of the research was to find out what children find easy and difficult about number questions.
Children were then presented with a partitioning problem which was characterised in the form of a vignette, accompanied by an illustration (see Figure 3.2). Children were introduced to a character called Jon, and told that he had bought some bananas. Children were told that Jon likes to come home and keep the bananas in the two bowls, and that Jon was confused because there were ―so many ways to put the bananas in the bowls‖. The interviewer explained that the aim was to try to help Jon by telling him all the different ways he could keep his bananas in the two bowls.
Figure 3.2: Image used in Study 2 to support problem understanding
3.2.4.1 Example problem
Before each of the two partitioning problems in each condition, the interviewer presented an example to help children understand the task demands and what constituted a valid solution. The interviewer would explain: “One day, Jon bought 3 bananas [interviewer shows image of 3 bananas]. Watch how I use [my head/these cubes] to help me find all the ways the three bananas could be in the two bowls.” In the Physical condition the interviewer placed three Unifix (2cm plastic cubes) on the table. In the No Materials condition, the interviewer
154 pointed to his head (the teacher of the class had explained how this prompt was used when children were being asked to solve problems mentally).
The interviewer would then identify the four ways to partition three in the following order: 3 & 0, 1 & 2, 2 & 1, and 0 & 3. In the Physical condition, the interviewer would partition the cubes before identifying the solution. Partitioning cubes involved moving the cubes into left and right groups in front of the interviewer. In the No Materials condition, the interviewer would simply point to the corresponding bowls when saying the verbal solutions. In the demonstration, the interviewer would explain that there could be “three in the red bowl and none in the green”, “one in the red bowls and two in the green”, “two in the red bowl and ...” On this third solution, the interviewer would purposefully pause and look at the child to prompt the child to say the solution (two in the green). If the child did not answer, the interviewer would use the image of the bananas and repeat “two in the red bowl and …” All children were able to complete this, as well as the final solution which again the interviewer prompted “and none in the red bowl and ….” (three in the green). The prompts for children to complete the solution were to ensure understanding and for children to practise giving numerical answers for each part.
3.2.4.2 Partitioning problem
After the demonstration problem, the interviewer removed the picture of three bananas but kept the picture of the stick figure and the two bowls. The children were then told that on another day Jon went shopping and bought 6/7 bananas. The order of total amount to partition was the same for all children: 6 followed by 7. Similarly to the example, in the Physical condition, children were presented with the correct total number of cubes to partition, which were placed in a line in front of the child. Children were then asked to use the cubes (use their heads in the No Materials condition) to tell the
155 interviewer all the ways Jon could put the 6/7 bananas in the two bowls. The children were reminded that, for each way, they were to say how many there were altogether in each bowl so that the interviewer could write down their answers. After solving the first partitioning problem, the interviewer would present the example and partitioning problem in the other condition. Condition order was counterbalanced between children.
3.2.4.3 Prompts during problem solving
For all problems, if children did not respond after 10 seconds they were prompted by the interviewer: “can you think of any ways that Jon can put the 6/7 bananas in the two bags?” For pauses after children had identified the first solution, the interviewer would prompt saying “is that all the ways or can you think of any more ways?” The session would end after two prompts had been given or if the child indicated that he/she had finished. If a child used non specific words such as ‗some‘ or ‗the rest‘ when identifying solutions, the interviewer would prompt by asking “so how many is „some‟/„the rest‟?
The interviewer wrote down all solutions given by the children so that they could see that their answers were being recorded (and that they were therefore important to the task) although they could not see what was actually being written down. Children would generally say or point to the bowl to which they were referring (e.g., “three in that one”) but if it was not clear the interviewer would prompt “three in which bowl?” The interviewer recorded the left bag as referring to the first part and right as the second.
For several children it was necessary to remind them of the need to identify partitions verbally in the Physical condition by stating the total amount, not to just show the configuration. Although it might be argued that this provided an unfair prompt for this condition, the prompt was a) only required for three children and b) only provided
156 once; if the child created a configuration and then looked to the interviewer, this was taken to mean that the child had created a solution12.